In reviewing theoretical and empirical aspects of the Multiplicative Error Model, the paper has two main sections, devoted to the univariate, respectively, multivariate versions of the \(\mathsf {MEM}\) . Its contribution can be synthesized as follows: we have proposed a modification of the standard specification of a \(\mathsf {MEM}\)  to highlight the composition of the conditional expectation of a positive variable of interest, in a way inspired by [1]} (\(\mathsf {SpMEM}\) ). From a theoretical point of view we have discussed how GMM applies to this model and the properties of the estimator, showing an equivalence with the Quasi Maximum Likelihood estimator under the Gamma assumption. In this respect, we have proceeded to an original set of applications in the univariate case on series of realized volatility from several markets, presenting estimation results both in tables and in the graphical decomposition achieved among the various components. Moreover, we have discussed the properties of estimated residuals graphically (correlogram and histograms), and more formally with distribution fitting tests. In the multivariate context, we have introduced the extension to the \(\mathsf {SpvMEM}\) model by [2]} to accommodate interdependence in the multivariate framework, showing, again, our preference for the GMM estimation strategy. The applications in the multivariate case relate to a trivariate model with absolute returns, realized volatility and implied volatility from option–based indices. Estimation results both for the \(\mathsf {vMEM}\)  and the \(\mathsf {SpvMEM}\) are complemented by some tests for HEAVY–type dynamics [3]} which show that the impact of absolute returns on realized volatility is relevant. Once again, attention is paid to the properties of estimated residuals, showing how substantially different their profile is in terms of underlying distributions, to the point that while a Gamma often fits the behavior for absolute returns, and a Beta\(^\prime \)  or a Log–Normal that of realized volatility, none of the distributions considered is suitable for implied volatility.
